.\" Automatically generated by Pod::Man 4.14 (Pod::Simple 3.40) .\" .\" Standard preamble: .\" ======================================================================== .de Sp \" Vertical space (when we can't use .PP) .if t .sp .5v .if n .sp .. .de Vb \" Begin verbatim text .ft CW .nf .ne \\$1 .. .de Ve \" End verbatim text .ft R .fi .. .\" Set up some character translations and predefined strings. \*(-- will .\" give an unbreakable dash, \*(PI will give pi, \*(L" will give a left .\" double quote, and \*(R" will give a right double quote. \*(C+ will .\" give a nicer C++. Capital omega is used to do unbreakable dashes and .\" therefore won't be available. \*(C` and \*(C' expand to `' in nroff, .\" nothing in troff, for use with C<>. .tr \(*W- .ds C+ C\v'-.1v'\h'-1p'\s-2+\h'-1p'+\s0\v'.1v'\h'-1p' .ie n \{\ . ds -- \(*W- . ds PI pi . if (\n(.H=4u)&(1m=24u) .ds -- \(*W\h'-12u'\(*W\h'-12u'-\" diablo 10 pitch . if (\n(.H=4u)&(1m=20u) .ds -- \(*W\h'-12u'\(*W\h'-8u'-\" diablo 12 pitch . ds L" "" . ds R" "" . ds C` "" . ds C' "" 'br\} .el\{\ . ds -- \|\(em\| . ds PI \(*p . ds L" `` . ds R" '' . ds C` . ds C' 'br\} .\" .\" Escape single quotes in literal strings from groff's Unicode transform. .ie \n(.g .ds Aq \(aq .el .ds Aq ' .\" .\" If the F register is >0, we'll generate index entries on stderr for .\" titles (.TH), headers (.SH), subsections (.SS), items (.Ip), and index .\" entries marked with X<> in POD. Of course, you'll have to process the .\" output yourself in some meaningful fashion. .\" .\" Avoid warning from groff about undefined register 'F'. .de IX .. .nr rF 0 .if \n(.g .if rF .nr rF 1 .if (\n(rF:(\n(.g==0)) \{\ . if \nF \{\ . de IX . tm Index:\\$1\t\\n%\t"\\$2" .. . if !\nF==2 \{\ . nr % 0 . nr F 2 . \} . \} .\} .rr rF .\" ======================================================================== .\" .IX Title "Math::PlanePath::SquareArms 3pm" .TH Math::PlanePath::SquareArms 3pm "2021-01-23" "perl v5.32.0" "User Contributed Perl Documentation" .\" For nroff, turn off justification. Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::PlanePath::SquareArms \-\- four spiral arms .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::SquareArms; \& my $path = Math::PlanePath::SquareArms\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path follows four spiral arms, each advancing successively, .PP .Vb 10 \& ...\-\-33\-\-29 3 \& | \& 26\-\-22\-\-18\-\-14\-\-10 25 2 \& | | | \& 30 11\-\- 7\-\- 3 6 21 1 \& | | | | \& ... 15 4 1 2 17 ... <\- Y=0 \& | | | | | \& 19 8 5\-\- 9\-\-13 32 \-1 \& | | | \& 23 12\-\-16\-\-20\-\-24\-\-28 \-2 \& | \& 27\-\-31\-\-... \-3 \& \& ^ ^ ^ ^ ^ ^ ^ \& \-3 \-2 \-1 X=0 1 2 3 ... .Ve .PP Each arm is quadratic, with each loop 128 longer than the preceding. The perfect squares fall in eight straight lines 4, with the even squares on the X and Y axes and the odd squares on the diagonals X=Y and X=\-Y. .IX Xref "Square numbers" .PP Some novel straight lines arise from numbers which are a repdigit in one or more bases (Sloane's A167782). \*(L"111\*(R" in various bases falls on straight lines. Numbers \*(L"[16][16][16]\*(R" in bases 17,19,21,etc are a horizontal at Y=3 because they're perfect squares, and \*(L"[64][64][64]\*(R" in base 65,66,etc go a vertically downwards from X=12,Y=\-266 similarly because they're squares. .PP Each arm is N=4*k+rem for a remainder rem=0,1,2,3, so sequences related to multiples of 4 or with a modulo 4 pattern may fall on particular arms. .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::SquareArms\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::SquareArms\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::SquareArms->new ()" Create and return a new path object. .ie n .IP """($x,$y) = $path\->n_to_xy ($n)""" 4 .el .IP "\f(CW($x,$y) = $path\->n_to_xy ($n)\fR" 4 .IX Item "($x,$y) = $path->n_to_xy ($n)" Return the X,Y coordinates of point number \f(CW$n\fR on the path. For \f(CW\*(C`$n < 1\*(C'\fR the return is an empty list, as the path starts at 1. .Sp Fractional \f(CW$n\fR gives a point on the line between \f(CW$n\fR and \f(CW\*(C`$n+4\*(C'\fR, that \&\f(CW\*(C`$n+4\*(C'\fR being the next point on the same spiralling arm. This is probably of limited use, but arises fairly naturally from the calculation. .SS "Descriptive Methods" .IX Subsection "Descriptive Methods" .ie n .IP """$arms = $path\->arms_count()""" 4 .el .IP "\f(CW$arms = $path\->arms_count()\fR" 4 .IX Item "$arms = $path->arms_count()" Return 4. .SH "FORMULAS" .IX Header "FORMULAS" .SS "Rectangle N Range" .IX Subsection "Rectangle N Range" Within a square X=\-d...+d, and Y=\-d...+d the biggest N is the end of the N=5 arm in that square, which is N=9, 25, 49, 81, etc, (2d+1)^2, in successive corners of the square. So for a rectangle find a surrounding d square, .PP .Vb 1 \& d = max(abs(x1),abs(y1),abs(x2),abs(y2)) .Ve .PP from which .PP .Vb 2 \& Nmax = (2*d+1)^2 \& = (4*d + 4)*d + 1 .Ve .PP This can be used for a minimum too by finding the smallest d covered by the rectangle. .PP .Vb 4 \& dlo = max (0, \& min(abs(y1),abs(y2)) if x=0 not covered \& min(abs(x1),abs(x2)) if y=0 not covered \& ) .Ve .PP from which the maximum of the preceding dlo\-1 square, .PP .Vb 4 \& Nlo = / 1 if dlo=0 \& \e (2*(dlo\-1)+1)^2 +1 if dlo!=0 \& = (2*dlo \- 1)^2 \& = (4*dlo \- 4)*dlo + 1 .Ve .PP For a tighter maximum, horizontally N increases to the left or right of the diagonal X=Y line (or X=Y+/\-1 line), which means one end or the other is the maximum. Similar vertically N increases above or below the off-diagonal X=\-Y so the top or bottom is the maximum. This means for a rectangle the biggest N is at one of the four corners, .PP .Vb 4 \& Nhi = max (xy_to_n (x1,y1), \& xy_to_n (x1,y2), \& xy_to_n (x2,y1), \& xy_to_n (x2,y2)) .Ve .PP The current code uses a dlo for Nlo and the corners for Nhi, which means the high is exact but the low is not. .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::DiamondArms, Math::PlanePath::HexArms, Math::PlanePath::SquareSpiral .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde .PP This file is part of Math-PlanePath. .PP Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the \s-1GNU\s0 General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. .PP Math-PlanePath is distributed in the hope that it will be useful, but \&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0 or \s-1FITNESS FOR A PARTICULAR PURPOSE.\s0 See the \s-1GNU\s0 General Public License for more details. .PP You should have received a copy of the \s-1GNU\s0 General Public License along with Math-PlanePath. If not, see .